PhD DISSERTATION DEFENSE

In this defense, we discuss various dependence structures in certain classes of Markov processes. Dependence structures like “association,” “supermodular dependence,” or “orthant dependence” are stronger forms of dependence than the classical notion of “correlation.” They carry very nice properties that make them useful in probability in statistics. In multivariate stochastic processes, these dependence structures can also help us describe the evolution of the processes over time.

The stochastic processes under study are Levy-type Markov processes. Such processes have behavior that is “locally infinitely divisible-like,” which not only gives it nice analytical properties, but also makes it useful in applications. We present a general characterization of positive dependence structures for time-homogeneous Levy-type Markov processes. In particular, we also characterize positive dependence for certain Levy-type jump processes. We then extended these results to the time-inhomogeneous case. Finally, we investigate negative dependence structures and give their characterizations in infinitely divisible random vectors and jump-Levy processes.

PhD DISSERTATION DEFENSE

Many unconditionally energy stable schemes for the physical models will lead to a highly nonlinear elliptic PDE systems which arise from time discretization of parabolic equations. In this defense, I will discuss two efficient and practical preconditioned solvers-- Preconditioned Steepest Descent (PSD) solver and Preconditioned Nonlinear Conjugate Gradient (PNCG) solver -- for the nonlinear elliptic PDE systems. The main idea of the preconditioned solvers is to use a linearized version of the nonlinear operator as a pre-conditioner, or in other words, as a metric for choosing the search direction. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. Numerical simulations for some important physical application problems -- including Epitaxial Thin Film equation with slope selection, Square Phase Field Crystal equation and Functionalized Cahn-Hilliard equation-- are carried out to verify the efficiency of the solvers.